The equilibrium distribution functions \(f_i^{eq}(\mathbf {x},t)\)  for an incompressible Lattice Boltzmann model [1]} and the forcing terms \(F_i(x,t)\)  [2]} are resepctively defined as
\(f_i^{eq}(x,t)= \frac{\omega _i P(x,t)}{c_s^2}+\omega _i\rho \left[\frac{e_i\cdot v(x,t)}{c_s^2}+\frac{(e_i\cdot v(x,t))^2}{2c_s^4}-\frac{v(x,t)^2}{2c_s^2} \right] \quad \text{and}\) 
\(F_i(x,t)=\left(1-\frac{1}{2\tau }\right)\times \omega _i\times \left[\left(\frac{{e_i-v(x,t)}}{c_s^2}+\frac{e_i\cdot v(x,t)}{c_s^4}e_i\right)\cdot b(x,t)\right],\) 
